设 $\left(X_{1}, Y_{1}\right),\left(X_{2}, Y_{2}\right), \ldots,\left(X_{n}, Y_{n}\right)$ 为来自总体 $N\left(\mu_{1}, \mu_{2} ; \sigma_{1}^{2}, \sigma_{2}^{2} ; \rho\right)$ 的简单随机样本, 令
$$
\theta=\mu_{1}-\mu_{2}, \bar{X}=\frac{1}{n} \sum_{i=1}^{n} X_{i}, \bar{Y}=\sum_{i=1}^{n} Y_{i}, \hat{\theta}=\bar{X}-\bar{Y},
$$
则 ( ) .
$\text{A.}$ $E(\hat{\theta})=\theta, D(\hat{\theta})=\frac{\sigma_{1}^{2}+\sigma_{2}^{2}}{n}$.
$\text{B.}$ $E(\hat{\theta})=\theta, D(\hat{\theta})=\frac{\sigma_{1}^{2}+\sigma_{2}^{2}-2 \rho \sigma_{1} \sigma_{2}}{n}$.
$\text{C.}$ $E(\hat{\theta}) \neq \theta, D(\hat{\theta})=\frac{\sigma_{1}^{2}+\sigma_{2}^{2}}{n}$
$\text{D.}$ $E(\hat{\theta}) \neq \theta, D(\hat{\theta})=\frac{\sigma_{1}^{2}+\sigma_{2}^{2}-2 \rho \sigma_{1} \sigma_{2}}{n}$